Fricke-Macbeath Curve

The Fricke-Macbeath curve, also called the Macbeath curve, is a smooth projective algebraic curve of genus 7. Its complex points form a compact Riemann surface, also called the Macbeath surface or Fricke-Macbeath surface. The conformal maps from the surface to itself form an automorphism group isomorphic to the projective special linear group , of group order . The curve is unique up to isomorphism among curves of genus 7 attaining this Hurwitz bound (Macbeath 1965, Top and Verschoor 2018).

An affine plane model over , attributed to Bradley Brock, is

The plane curve has 14 ordinary double points, and its projective closure has no singularities at infinity. Its normalization is the smooth Fricke-Macbeath curve (Top and Verschoor 2018).

The curve carries the Hurwitz map . The 1-skeleton of this map is the Fricke-Macbeath graph (Bokowski and Cuntz 2018).

Fricke introduced the associated Riemann surface in 1899, and Macbeath presented explicit algebraic equations in 1965 (Fricke 1899, Macbeath 1965, Top and Verschoor 2018).